SuStel:

jatlhqu'meH tlhIngan Hol naQ law' Quenya naQ puS

jIH:

Or is it rather "in order that he/they speaks/speak a lot, klingon is more complete than quenya" ?

SuStel

My intention was is the last one

Perhaps the reason of my confusion, becomes clearer now. If instead of
{jatlhqu'meH tlhIngan Hol naQ law' Quenya naQ puS}, we had
{jatlhqu'lu'meH tlhIngan Hol naQ law' Quenya naQ puS} meaning "in
order for someone to speak..", then I could have understood the
meaning better. Reading the {jatlhqu'meH tlhIngan Hol naQ law' Quenya
naQ puS} and understanding "in order that he/they speaks/speak a lot,
klingon is more complete than quenya", I begun to wonder who the
"he/they" was/were. Let alone that I did the mistake of thinking that
the {tlhIngan Hol} was part of the {meH}ed construction, as opposed to
the law'/puS construction.

I'm not sure that would have helped. You weren't interpreting tlhIngan Hol as the subject of jatlhqu'meH; you were interpreting it as the head noun of jatlhqu'meH. Adding a -lu' wouldn't have changed anything.

Klingon purpose clauses are often used in a sort of infinite way. You don't say ghojlu'meH taj; you say ghojmeH taj. A subject is not always necessary or even implied. Sometimes it is speculated that you need a subject if the purpose clause attaches to a sentence instead of a noun, but we don't really know, and no survey of canon has been done recently on that.

-- 
SuStel
http://trimboli.name